All eigenstates and eigenvalues are determined for the spin- 1/2 $XXZ$ chain $H = 2J sum_i ( S_{i}^{x} S_{i + 1}^{x} + S_{i}^{y} S_{i + 1}^{y} + Delta S_i^z S_{i + 1}^{z})$ for rings with up to N=16 spins, for anisotropies $Delta=0 , cos(0.3pi)$, and 1. The dynamic spin pair correlations $< S_{l+n}^{mu}(t) S_l^{mu} > , (mu=x,z)$, the dynamic structure factors $S^{mu}(q,omega)$, and the intermediate structure factors $I^{mu}(q,t)$ are calculated for arbitrary temperature T. It is found, that for all T, $S^{z}(q,omega)$ is mainly concentrated on the region $|omega| < varepsilon_2(q)$, where $varepsilon_2(q)$ is the upper boundary of the two-spinon continuum, although excited states corresponding to a much broader frequency spectrum contribute. This is also true for the Haldane-Shastry model and the frustrated Heisenberg model. The intermediate structure factors $I^{mu}(q,t)$ for $Delta eq 0$ show exponential decay for high T and large q. Within the accessible time range, the time-dependent spin correlation functions do not display the long-time signatures of spin diffusion.
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